[EM] Thoughts on majority potential simulations

[Anthony Simmons]

>> From:  Richard Moore
>> Subject: [EM] thoughts on majority potential simulations

[...]

>> 4. I am going to use a 2-dimensional policy space. I am
>> going to calculate utilities based on the L1 (Hamming)
>> distance between the voter and the candidate. The
>> rationale for L1 is that if policy X has a cost to the
>> voter and policy Y has a cost to the voter (not
>> necessarily monetary costs but such can be used as a
>> tangible example), then the combined costs of both
>> policies is a simple sum. If there are any arguments for
>> using L2 distance please let me hear them.

Usually, if there are a whole lot of factors (as in "factor
analysis"), they aren't independent.  For example, you'd
imagine that if I'm morally opposed to ice cream, I'd most
likely be opposed to frozen yogurt as well.  If you make one
the X coordinate and one the Y coordinate, and plot the
positions of actual moral philosophers, you'd expect to find
a pretty steady correlation between X and Y, with most of the
pack along a straight line through the origin and
representing a third factor (a derived one in this case), Z,
"moral correctness of frozen confections".

It's reasonable to consider Z as basic as X and Y, so we'd
like to be able to rotate our graph so that this factor is
horizontal or vertical and becomes one of the coordinate
axes, without changing any of the relationships between the
variables or the distances between points on the Z line; we
wouldn't want a policy to become more or less extreme on our
scale just because we rotated the scales.

Using the root-sum-of-squares distance makes all of this very
clean.  Of course, you could preserve distances in other
ways.  If you were using the the city block distance, and you
rotated the coordinates so that (1, 1) ended up on the X
axis, you could stretch it to (2, 0).  After all, if you're
not using Euclidean distance, then there's no requirement
that (1, 1) rotates onto (1.414, 0).  But when you throw out
the way we normally measure distances, you throw out the
underlying geometry, and we all know that the most important
thing about any measurement is the underlying geometric
aesthetics.

Another consideration:  In the illustration above, the data
actually lies along one dimension, embedded in a two-
dimensional space.  If we were to add popsicles to ice cream
and yogurt, we would appear to have a third dimension, but
it's spurious.  The data still lie along a single dimension.

In real life, this isn't likely to be so clear.  We'd expect
some moral philosophers to adhere to the orthodox position
that petty distinctions between frozen sweets are the work of
the devil, while others might take a more complex view.
Thus, as the number of frozen confections increases, the
number of dimensions required to describe the actual data
would increase, but not as fast as the number of raw
dimensions.  For example, with five desserts, perhaps we
could describe all positions as lying on a two-dimensional
surface within the five-dimensional ambient space.

Using root-sum-of-squares reflects this situation nicely.
If we want to know how far from neutral the position of
treating all desserts equally is (that is, how far from the
origin is the point (1, 1, ...., 1), we find that adding more
desserts has less and less effect.  Which is what we'd want.
We'd like the difference between plots of one and two
desserts to be large, but the difference between 72 and 73 to
be almost identical.  Simply creating an artificial
disctinction between chocoloate and vanilla ice cream
shouldn't change my fundamental position if all of the
philosophers consider them to be of equal moral merit.

Just some thoughts.  Sorry it's so long and rambling.  I
didn't have time to make it shorter.

    Tony

P.S.:  I just ran across this quote.  It's from the
bibliography in "Loring Ensemble Rules", fetchable from
http://member.aol.com/loringrbt/

    "Dimensional Analysis of Ranking Data". by Henry E. Brady
    American Journal of Political Science. 34 (11/90).  Brady
    found that a spatial model with 2 or 3 opinion dimensions
    depicted a real electorate well.  Knowing a voter's
    position on 1 issue helps predict his positions on
    related issues in that dimension.  Knowing his 3D
    position helps predict his ranking of candidates.

I haven't read the paper.